# Equilibrium with substitute goods

I am attempting to solve the following problem

The demand functions for two substitute goods, the production cost of which equals $$c_1$$ and $$c_2$$, are $$q_1 = a_1 + b_{11}p_1 + b_{12}p_2$$ and $$q_2 = a_2 + b_{21}p_1 + b_{22}p_2$$.

How can I find equilibrium under simultaneous and sequential interaction in a price oligopoly

(the values of the coefficients $$a_1,a_2,b_{11} ...$$ can be selected by yourself. Although, if I'm not mistaken, the sign matter)

I need help in understanding the difference between simultanious and sequential interaction solution

• @HerrK, first of all I have to figure out the condition for substitute goods Apr 24, 2021 at 11:11
• What is a price oligopoly? Apr 24, 2021 at 11:51
• Firms choose quantities? Apr 24, 2021 at 12:15
• That gives you the simultaneous version. For the Otter, the second one to player faces the same problem. You can substitute the solutution into the maximization problem of the first. But you should have seen this before. Apr 24, 2021 at 20:32

Let's start by writing down the profit functions: $$\pi_1 = p_1 q_1 - c_1 q_1 = (p_1 - c_1) q_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2)$$ and similarlily: $$\pi_2 = (p_2 - c_2)(a_2 + b_{21} p_1 +b_{22} p_2)$$

In a simultaneous equilibrium, both firms simultaneously maximize their profits taking the strategies of the players as given.

Doing this will give you a best response of player 1 that will be a function of $$p_2$$: $$p_1 = p_1^\ast(p_2)$$. Similarly, the best response of player 2 will be a function of $$p_1$$: $$p_2 = p_2^\ast(p_1)$$. In a Nash equilibrium, you are looking for a solution for both equations simultaneously: $$p_1 = p_1^\ast(p_2),\\ p_2 = p_2^\ast(p_1).$$

In a sequential equilibrium on the other hand, you assume that one firm, say 1, chooses $$p_1$$ first and next firm 2 decides in $$q_2$$, after being informed about the value of $$q_1$$ chosen by firm 1. Firm 1, however, is smart enough, such that it can predict the optimal action that firm 2 is going to take in stage 2.

As such, we solve the game by backward induction, first solving for firm 2, giving $$p_2 = p_2^\ast(p_1)$$ and then we substitute this into the profit function of firm 1. $$\pi_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2^\ast(p_1))$$ This substitution conforms to the fact that player 1 knows that by choosing $$p_1$$ it will influence the price of firm 2. So firm 1 takes the future behaviour of firm 2 into account when choosing its optimal level of $$q_1$$.

Maximizing $$\pi_1$$ with respect to $$p_1$$ will give an optimal value $$p_1^\ast$$ (which will no longer be a function of $$p_2$$).

Then, to compute the equilibrium, you substitute this value into the best response for firm 2: $$p_2 = p_2^\ast(p_1^\ast).$$